On 3/25/2013 7:07 AM, WM wrote:> On 25 Mrz., 00:49, Virgil <vir...@ligriv.com> wrote:>> In article>> <39dd320b-1f56-4cf7-bb03-f0f634420...@l5g2000yqe.googlegroups.com>,>>>> WM <mueck...@rz.fh-augsburg.de> wrote:>>> On 24 Mrz., 20:39, Virgil <vir...@ligriv.com> wrote:>>>>>> But if David had left out the world "all", and said merely>>>> "In fact, Aleph_0 lines are required>>>> (necessary sufficient) to contain all of the naturals.">>>> then David would have been correct, since EVERY set of aleph_0 lines is>>>> sufficient but no set of less than aleph_0 lines is sufficient.>>>>> We know your statements of faith. But where do you get aleph_0 lines>>> without using lines of the infinite set of aleph_0 lines that, as>>> provable in mathematics, are not sufficient?>>>> Which infinite sets of lines does WM claim are provably not sufficient?>> All FISONs are not sufficient, because forall F in the set of FISONs:> There are infinitely many natural numbers not covered by F and all its> predecessors and all its followers.>

That is why 'all' means something different from 'one'.

That is the genius of what happened in the nineteenth century.Mathematics was seen to be a form of logic that could refer toparts of individuals as individuals in contrast to the classicallogic that preceded that period.

>> THEOREM: To have a subset of the infinite set of lines(FISONs) whose>> union is |N, it is both necessary and sufficient that that subset of>> lines also be infinite.>> Nonsense. All FISONs cannot be sufficient, since no FISON is> necessary.>> Corollary: To catch a unicorn it is both necessary and sufficient to> ask an infinity of horses to help.